Causal Inference for Observational Longitudinal Survival Data

Mark Bech Knudsen

Context

Thesis written in blocks 1 & 2 autumn 2022, supervised by Torben Martinussen & Helene Rytgaard @ Biostatistics KU.

Thesis explores 3 topics relating to causal inference for non-randomized survival data:

  • Do hazard ratio estimates from the Cox model have a causal interpretation for a time-dependent treatment?
  • A problem with Markov models for survival data with time-dependent covariates.

  • Causal inference in continuous-time using inverse probability of treatment weights.

Defining causal effects

Assume a binary treatment \(A \in \{0, 1\}\) that is given at time \(t = 0\). Wish to infer the effect of treatment on survival.

For each person, suppose there exists two potential outcomes \(T^0\) and \(T^1\):

  • \(T^0\) is the survival time if the person does not receive treatment.
  • \(T^1\) is the survival time if the person receives treatment.

The effect of treatment could then be defined as \[ P(T^1 > t) - P(T^0 > t) \] i.e. how much more likely is it for the person to survive if they receive treatment vs. if they do not?

Or on the population scale: What would the survival rate be if we treated everyone vs. if we did not treat anyone?

Different from the conditional association \(P(T > t \mid A = 1) - P(T > t \mid A = 0)\).

Causal vs. conditional effects

Randomized treatment

Assume a randomized treatment \(A\). Then the subpopulation with \(A = a\) is representative of what would have happened to the rest of the population, had they been treated.

Scale up subpopulation to full population size with weights \[ W_i^a = \frac{1(A_i = a)}{P(A = a)} \]

An estimate of the probability of survival, had everyone received treatment \(a\): \[ P(T^a > t) = \frac{1}{n} \sum_{i=1}^n W_i^a 1(T_i > t) \]

The \(W_i^a\) are referred to as inverse probability of treatment weights (IPTW).

Confounding

Suppose now that a confounder \(X \in \{0, 1\}\) (low/high risk) affects both treatment and outcome (non-randomized data).

If \(X\) is the only confounder, then we still have representativity within levels of \(X\). So we can pretend randomized treatment within levels of \(X\).

So we assign different weights depending on \(X\): \[ W_i^a = \frac{1(A_i = a)}{P(A = a \mid X_i)} \]

This works regardless of the dimension of \(X\), and also for continuous covariates.

Time-varying treatment

Consider a case where treatment is a binary counting process \(A(t)\) that starts at 0 and jumps to 1 at time \(\tau\) when a treatment is undertaken (e.g. surgery).

Covariates \(X(t)\) are also time-dependent.

Can generalize to a weight process \[ W_i(t) = \frac{1}{\lambda^A(\tau \mid \mathcal{F}_{\tau-})^{A(t)} \mathrm{e}^{-\Lambda^A(t \wedge \tau)}} \] where \(\lambda^A\) is the hazard/rate of treatment initiation and \[ \Lambda^A(t) = \int_0^t \lambda^A(s \mid \mathcal{F}_{s-}) \mathrm{d} s \]

The processes \(W_i(t)\) are quite nice (exponential martingales), and can be estimated from data.

A data example

Observational data (\(n = 601\)) from oncologists at Rigshospitalet.

  • \(T\) time from diagnosis of incurable throat cancer to death.
  • \(A(t)\) insertion of metalic tube (stent) into throat.
  • A bunch of covariates \(X(t)\), mostly measured at baseline.

Weighting does not change the conclusion much. But we did not have enough information about time-dependent confounders.

Thank you!

And good luck when you write your thesis.